At the beginning of her lesson, Molly McNinch asks her students to talk to a peer and describe what it means to “model a problem.” Her students share their ideas: modeling a problem means making a diagram, a graph, a chart, a table. Molly challenges her students to “think about ways that you can organize your data that would be beneficial to getting to the solution.” She displays videos of different rolling cups that her students had watched previously, with the purpose of understanding how the diameter of the cup and the roll radius relate to each other, and how different objects behave differently.

9th Grade Math - Modeling through Geometry: Circumference of a Cup’s Roll*Molly McNinch, San Lorenzo Unified School District, Woodside High School, Woodside, California*

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MOLLY MCNINCH: Happy Friday. So … last class, we were talking about our rolling cups. Okay. So our objective is to produce mathematical solutions to modeling the Rolling Cups problem. Now what we did yesterday, we watched riveting videos of the rolling cups. So one of the things that I want to kind of highlight here is this term modeling. Okay, so -- this term modeling, I want you guys to take about 30 seconds, share with the people at your table what you think it is “to model" something. Now we're not talking about clothes, but like model a problem. What is the first word that comes to your mind? Maybe the first word that comes to my mind when I think modeling... First word I think is “America's Next Top Model,” but when I'm relating it to math, I think like maybe like a picture of something. Take 30 seconds, talk to your group, and then we're going to come back and kind of put up all the words we think of when we think of modeling. 30 seconds. Go.

Okay, so -- modeling. What did we come up with? Let's go to this table right here. What was a word you came up with? Let's go, Gavin. “Solution.” That works. Cool. “Solution.” Let's go to Nathan.

STUDENT: We came up with a diagram with a bunch of numbers and equations on it.

MOLLY MCNINCH: Okay. “Diagram with numbers, equations.” Let's go to Julian.

STUDENT: We also said diagram, but like pictures on it.

MOLLY MCNINCH: Okay. “Pictures.” All right, so -- Kelsey.

STUDENT: “Figure.”

MOLLY MCNINCH: Figure. Yeah, so remember what we talked about when we were referring to transformations? The figure is the first one. The image is what you transform.

Let's go, Keilty.

STUDENT: A graph.

MOLLY MCNINCH: A graph, yes. A graph is definitely a form of modeling. And let's go here, Danny.

STUDENT: A chart.

MOLLY MCNINCH: A chart. Yes. So all of these are ways that you can model mathematics. Now one of the things, it's not explicitly stated here, but it kind of goes with chart and diagram, is a table. So a table -- I think tables are very underused. So when you guys are doing this problem today, I want you to think about ways that you could organize your data that would be beneficial to getting to the solution.

Now, I'm passing out the pink papers, and these are not yours ... I'm passing out the pink papers, which have information. I gave you feedback. A lot of you guys had really good ideas about how the two diameters related to each other, and so you were kind of approaching that.

So some of you guys got really far with your thinking. What I want you to focus on today is: how can you relate the two diameters to the slant height, and then how can you kind of think about the relationship between all three measurements?

So I'm going to give you guys -- so we're going to rewatch the videos. Whoops, I need my pen there. Let's rewatch the videos. Going to load it over here. Now, remember we had those four cups? Okay. So -- and the soup can. One, two, three, four. Take 5 seconds. Decide which one you think is going to have the largest roll radius. One, two, three, or four. Take about 5 seconds. All right, give me a number of fingers. Where are you guys at, one, two, three, or four?

All right, you all voted for three. I want you to look at the differences between the cups and the can. All right. So -- short glass. This is, I like to refer to it as, the short and stout. Okay, so short and stout gave us ... What is our roll radius there? Okay, 26 and a quarter inches. Again, this is going to refer [to] that first cup on your table, A. So this is Cup A, our plastic cup, also known as “the IKEA cup.” I feel like they do sell these at IKEA. Maybe Bed Bath & Beyond, who knows. I mean, this one makes the most circles, and it's the most colorful.

Okay, so this one was smaller, not as stout as the other one, and it had a smaller roll radius. Now the tall glass, this is the one that most everyone voted for, and everyone in third period voted for this glass. Okay, so that's the largest ... Yeah? Go ahead.

STUDENT: Put it here?

MOLLY MCNINCH: Yeah. Let's do that. I'm miked, you guys. Yeah. It's my only chance to be famous. Okay, anyways, and -- so this was our tall glass, the one everyone voted for, and then this is our soup can. Okay. Let's watch it. Okay. Now this -- look at the difference between the bottom, so the narrow rate -- diameter and the wide diameter, the difference between them if we look ... Can I go back to the front? So the diameters are very similar. In fact, they are the same. So how does that affect the roll radius? Now, as I'm passing these back, I want you guys to ... okay, so, as we're passing these back, oh no ... Smart Board ...okay.

I try to honor that they're children still. Because I work primarily with freshmen, really. I have 13 sophomores out of all of my students. I try to keep in mind they're still young, but ... When I start to get down on myself about teaching, I try to remind myself I do this because I do care. If you ask my students to describe my classroom, they’ll say, "Yeah, it's pretty chill." Because it's mellow and they're small classes and they move at a pretty comfortable pace. My Geometry classes, we have a lot of fun.... So if a new student came into my Geometry class they would probably hear a lot of, "It's pretty mellow in here, but the work's really hard." Which is exactly what I want. I want it to be a safe environment for them, but you have to work.